Erdős–Dushnik–Miller theorem

Erdős–Dushnik–Miller theorem Not to be confused with Dushnik–Miller theorem.

In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph.[1] The theorem was first published by Ben Dushnik and E. w. Miller (1941), in both the form stated above and an equivalent complementary form: every infinite graph contains either a countably infinite clique or an independent set with equal cardinality to the whole graph. In their paper, they credited Paul Erdős with assistance in its proof. They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order.[2] The same theorem can also be stated as a result in set theory, using the arrow notation of Erdős & Rado (1956), come {displaystyle kappa rightarrow (kappa ,aleph _{0})^{2}} . Ciò significa che, for every set {stile di visualizzazione S} of cardinality {displaystyle kappa } , and every partition of the ordered pairs of elements of {stile di visualizzazione S} into two subsets {stile di visualizzazione P_{1}} e {stile di visualizzazione P_{2}} , there exists either a subset {stile di visualizzazione S_{1}subset S} of cardinality {displaystyle kappa } or a subset {stile di visualizzazione S_{2}subset S} of cardinality {stile di visualizzazione aleph _{0}} , such that all pairs of elements of {stile di visualizzazione S_{io}} belong to {stile di visualizzazione P_{io}} .[3] Qui, {stile di visualizzazione P_{1}} can be interpreted as the edges of a graph having {stile di visualizzazione S} as its vertex set, in quale {stile di visualizzazione S_{1}} (if it exists) is a clique of cardinality {displaystyle kappa } , e {stile di visualizzazione S_{2}} (if it exists) is a countably infinite independent set.[1] Se {stile di visualizzazione S} is taken to be the cardinal number {displaystyle kappa } si, the theorem can be formulated in terms of ordinal numbers with the notation {displaystyle kappa rightarrow (kappa ,omega )^{2}} , intendendo che {stile di visualizzazione S_{2}} (when it exists) has order type {stile di visualizzazione omega } . For uncountable regular cardinals {displaystyle kappa } (and some other cardinals) this can be strengthened to {displaystyle kappa rightarrow (kappa ,omega +1)^{2}} ;[4] però, it is consistent that this strengthening does not hold for the cardinality of the continuum.[5] The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem, and Paul Erdős's first significant result in set theory.[6] Riferimenti ^ Salta su: a b Milner, e. C.; Pouzet, M. (1985), "The Erdős–Dushnik–Miller theorem for topological graphs and orders", Order, 1 (3): 249–257, doi:10.1007/BF00383601, SIG 0779390, S2CID 123272176; see in particular Theorem 44 ^ Dushnik, Ben; Miller, e. w. (1941), "Partially ordered sets", Giornale americano di matematica, 63 (3): 600–610, doi:10.2307/2371374, JSTOR 2371374, SIG 0004862; see in particular Theorems 5.22 e 5.23 ^ Foresta, Paolo; Rado, R. (1956), "A partition calculus in set theory", Bollettino dell'American Mathematical Society, 62 (5): 427–489, doi:10.1090/S0002-9904-1956-10036-0, SIG 0081864 ^ Shelah, Sahara (2009), "The Erdős–Rado arrow for singular cardinals", Bollettino matematico canadese, 52 (1): 127–131, doi:10.4153/CMB-2009-015-8, SIG 2494318 ^ Shelah, Sahara; Stanley, Lee J. (2000), "Filters, Cohen sets and consistent extensions of the Erdős–Dushnik–Miller theorem", Il diario della logica simbolica, 65 (1): 259–271, arXiv:math/9709228, doi:10.2307/2586535, JSTOR 2586535, SIG 1782118, S2CID 2763013 ^ Hajnal, András (1997), "Paul Erdős' set theory", The mathematics of Paul Erdős, II, Algorithms and Combinatorics, vol. 14, Berlino: Springer, pp. 352–393, doi:10.1007/978-3-642-60406-5_33, SIG 1425228; see in particular Section 3, "Infinite Ramsey theory – early papers", p. 353 Categorie: Order theoryRamsey theorySet theory